Step 1

Heat 2 tablespoons olive oil in a 12-inch nonstick skillet over medium-high heat. Add shallots; cook until deep golden, stirring occasionally, 4 to 5 minutes. Drain shallots on paper towels, reserving drippings in skillet.

Step 2

Add green beans, water, 1/4 teaspoon salt, and 1/8 teaspoon pepper to reserved drippings. Cook over medium-high heat, covered, for 2 minutes. Cook, uncovered, until beans are crisp-tender and starting to blister, stirring occasionally, about 6 minutes more. Transfer to a platter; keep warm.

Step 3

For mushroom sauce, add the remaining 1 tablespoon oil to the skillet. Add mushrooms; cook and stir over medium heat for 1 minute. Add thyme, garlic, and remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Cook until mushrooms are golden, stirring frequently, about 2 minutes more. Mix in flour; gradually stir in milk. Cook and stir until thick and bubbly. Continue cooking and stirring for 2 minutes more.

Step 4

Pour mushroom sauce over beans and top with sauteed shallots and crumbled goat cheese.